Geodesic
Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 2, pp. 217-228

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Studying an operator equation $Au=f$ in Hilbert spaces one usually needs the adjoint operator $A^\star$ for $A$. Solving the ill-posed Cauchy problem for Dirac type systems in the Lebesgue spaces by an iteration method we propose to construct the corresponding adjoint operator with the use of normally solvable mixed problem for Helmholtz Equation. This leads to the description of necessary and sufficient solvability conditions for the Cauchy Problem and formulae for its exact and approximate solutions.
Keywords: mixed problems, Helmholtz equation, Dirac operators, ill-posed Cauchy problem. 
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     title = {Boundary problems for {Helmholtz} equation and the {Cauchy} problem for {Dirac} operators},
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Alexander A. Shlapunov. Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 4 (2011) no. 2, pp. 217-228. http://geodesic.mathdoc.fr/item/JSFU_2011_4_2_a9/